Page 167 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

The application of CF in genetic evaluation involves defining an equivalent model
using Eqn 9.8. For instance, using the example of the body weight of beef cattle,
assume that the multivariate model for observations measured on one animal is:

y = Xb + a + e
where y, X, b, a and e are vectors defined as in Eqn 5.1 with i = t, with var(a) = G ˘ and
var(e) = R. Assuming a CF has also been fitted for the covariance matrix for environ-
mental effects with a term included to account for measurement error, then:
R = FC F′ + Is e
2
p
where C contains the coefficient matrix associated with the CF for pe and variance
p
2
e is Is e. Using this equation and Eqn 9.8, an equivalent model to the multivariate
model can be written as:
y = Xb + Fu + Fpe + e
where u and pe are now vectors of random regression coefficients for random animal
and pe effects. Then var(u) = FCF′ and var(pe) = FC F′. The application of the
p
above model in genetic evaluation is illustrated in Example 9.2. Thus the breeding
value a for any time n can be calculated as:
n
k−1
n ∑
a = f i n i
t ()u
i=0
where f(t ) is the vector of Legendre polynomial coefficients evaluated at age t .
n n
Thus with a full order fit, the covariance function model is exactly equivalent to
the multivariate model. However, in practice, the order of fit is chosen such that
the estimated covariance matrix can be appropriately fitted with as few parameters
as possible. In the next section, the fitting of a reduced-order CF is discussed.

9.4.1 Fitting a reduced order covariance function

Equation 9.8 and the illustration given in the above section assumed a full-order poly-
nomial fit of G (k = t). Therefore, it was possible to get an inverse of F and hence
estimate C. However, for a reduced-order (k < t) fit, F has only k columns and a
direct inverse may not be possible. With the reduced fit, the number of coefficients to
be estimated are reduced to k(k + 1)/2. This is particularly important for large L, such
as test day milk yield within a lactation with t equal to 10 or 305 assuming monthly
or daily sampling and requiring t(t + 1)/2 coefficients to be estimated. Thus a reduced
order fit with k substantially lower than t could be very beneficial.
Kirkpartrick et al. (1990) proposed weighted least squares as an efficient method
ˇ
of obtaining an estimate of the reduced coefficient matrix (C) from the linear function
˘
of the elements of G. They outlined the following steps for the weighted least-square
˘
procedure. The procedure is illustrated using the example G for the body weight in
beef cattle given earlier, fitting polynomials of order one, i.e. only the first two
2
Legendre polynomials are fitted, thus k = 2. Initially, a vector g˘ of order t is formed
˘
by stacking the successive columns of G. Thus:
˘
˘
˘
˘
˘
˘
g′ = [G ,...,G , G ,...,G , G ,...,G ]
˘
11 n1 12 n2 1n nn
Analysis of Longitudinal Data 151